Definition 2.9 (\(\alpha \) -geodesic [1]})

The \(\alpha \) -geodesic connecting two probability vectors \(p(\mathbf {x})\)  and \(q(\mathbf {x})\)  is defined as
\(r_i(t) = c(t)f^{-1}_{\alpha }\Big \lbrace (1-t)f_{\alpha }(p(x_i)) + t f_\alpha (q(x_i))\Big \rbrace , \quad t \in [0,1]\) 
